What is Roots of equations: Definition and 12 Discussions
In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a set of n points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plane that can be deduced from the degree and the coefficients of the polynomial.
Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots, which define a disk containing all roots, or lower bounds on the distance between two roots. Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity
Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree n with real coefficients, which is less than
1
+
2
π
ln
(
n
)
{\displaystyle 1+{\frac {2}{\pi }}\ln(n)}
for n sufficiently large.
In this article, a polynomial that is considered is always denoted
p
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
,
{\displaystyle p(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n},}
where
a
0
,
…
,
a
n
{\displaystyle a_{0},\dots ,a_{n}}
are real or complex numbers and
a
n
≠
0
{\displaystyle a_{n}\neq 0}
; thus n is the degree of the polynomial.
On simplifying the given equation we get, x^2-x-1=0 and using the quadratic formula we get x=(1+√5)/2 and x=(1-√5)/2
Now, as the formula suggests, there are two possible values for x which satisfies the given equation.
But now, if we follow a process in any general calculator by entering...
Given : The quadratic equation ##x^2+px+q = 0## with coefficients ##p,q \in \mathbb{Z}##, that is positive or negative integers. Also the roots of the equation ##\alpha, \beta \in \mathbb{Q}##, that is they are rational numbers. To prove that ##\boxed{\alpha,\beta \in \mathbb{Z}}##, i.e. the...
It is given that ##x_1, x_2\; \text{and}\; x_3## are roots of the equation ##ax^2+bx+c=0##, which are pairwise distinct.
If indeed they are roots, we should have ##ax_1^2+bx_1+c= 0 = ax_2^2+bx_2+c= 0 = ax_3^2+bx_3+c= 0##.
On subtracting the first two, we obtain ##a(x_1^2-x_2^2)+b(x_1-x_2) =...
Mentor note: Thread moved to Diff. Equations subforum
Hello, few days ago I had a calculus test in which I had to find the general solution for the next differential equation: (D^8 - 2D^4 + D)y = 0.
"D" stands for the differential "Dy/Dx". However I could only find 2 of the roots on the...
For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?
Homework Statement
Let ##a,b,c## be positive integers and consider all the quadratic equations of the form ##ax^2-bx+c=0## which have two distinct real roots in ##(0,1)##. Find the least positive integers ##a## and ##b## for which such a quadratic equation exist.
Homework EquationsThe Attempt...
Is there a good way to relate the symmetries of the graphs of polynomials to the roots of equations?
There's lots of material on the web about teaching students how to determine if the graph of a function has a symmetry of some sort, but, aside from the task of drawing the graph, I don't find...
$a=1,2,3,4,5,------2011$, the roots of the equations $x^2-2x-a^2-a=0,$ are :
$(\alpha_1,\beta_1),(\alpha_2,\beta_2),----------,(\alpha_{2011},\beta_{2011})$ respectively
please find :
$\sum_{n=1}^{2011}(\dfrac{1}{\alpha_n}+\dfrac {1}{\beta_n})$
Hi,
I know what roots are and how to find them but I don’t know why they are so important.
What is that makes the points where a function become zero so important? I saw a similar post on this topic, but it talks about roots from an optimization point of view. However, finding roots...
Homework Statement
Hello everyone. My task is to find the largest positive root in a specific interval of a function using the bisection method, Newton-Raphson method, and secant method. I've written code for all three of these methods, but the only way I can find all of the roots is to hard...
Homework Statement
Hello everyone. My task is to find the largest positive root in a specific interval of a function using the bisection method, Newton-Raphson method, and secant method. I've written code for all three of these methods, but the only way I can find all of the roots is to hard...